From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6150 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Illusion and Forthrightness in Wikipedia Date: Sun, 12 Sep 2010 23:23:11 -0700 Message-ID: References: <730oimakP9232S03.1284336675@web03.cms.usa.net> Reply-To: Vaughan Pratt NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1284402514 20326 80.91.229.12 (13 Sep 2010 18:28:34 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 13 Sep 2010 18:28:34 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Mon Sep 13 20:28:32 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OvDlY-0006Vs-FF for gsmc-categories@m.gmane.org; Mon, 13 Sep 2010 20:28:32 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:37231) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OvDkJ-0006Rs-HB; Mon, 13 Sep 2010 15:27:15 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OvDkF-0000Bt-BI for categories-list@mlist.mta.ca; Mon, 13 Sep 2010 15:27:11 -0300 In-Reply-To: <730oimakP9232S03.1284336675@web03.cms.usa.net> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6150 Archived-At: On 9/12/2010 5:11 PM, Fred E.J. Linton wrote: > The knot? -- the familiar trefoil knot, aka (2,3)-torus knot. > The offending Wikipedia page? -- http://en.wikipedia.org/wiki/Torus_knot . > The problem? > [...] > if one uses PostScript to "draw" the curve with the parametrization given > above (with p=2 and q=3, of course), the result is the "lumpy" figure I've > put up, as .ps and .png files, respectively, here: > > http://tlvp.net/~tlvp/Trefoil/Wiki-2-3-Torus.ps , > http://tlvp.net/~tlvp/Trefoil/Wiki-2-3-Torus.png . There are really two trefoil knots: (i) the (2,3)-torus knot, which is a creature of topology, being defined up to homeomorphism of the complementary space, for which lumpiness is not an invariant; (ii) The trefoil arising as a a common motif in iconography and the visual arts. Thare are a number of variants of these, each defined up to similarity. They are dealt with in the section "Trefoils in religion and culture" on the Wikipedia page http://en.wikipedia.org/wiki/Trefoil_knot You seem to be interested in the second. Conceivably you could try modifying the topology article, but you will have to deal somehow with the objection that "lumpiness" is not a topological invariant, so why spoil a perfectly good parametrization by replacing it with a more complicated one that's solving a non-problem for this article? A better approach might be to expand the art section with a little mathematics giving a suitable parameterization of the kind you want. The ellipse-based one you suggest seems to work fine, and translates back from your postscript if I'm not mistaken as x = sin(3t)cos(t) + 3 cos(3t)sin(t) y = 3 cos(3t)cos(t) - sin(3t)sin(t) Here's another I came up with just now that also works. r = 4 - cos(3t) x = r sin(2t) - 3 sin(t) y = r cos(2t) + 3 cos(t) The parametrization in the torus knot article is this one with "4 - " replaced by "2 + " and the second half deleted from the expressions for x and y. Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]